matter where player I hides at least one line from the cover will find him so

# Matter where player i hides at least one line from

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matter where player I hides, at least one line from the cover will find him, so he is found with probability at least 1 /k . Thus K¨ onig’s lemma shows that this is, in fact, a joint optimal strategy, and that the value of the game is k 1 , where k is the size of the maximal set of independent 1’s. 2.7 General hide-and-seek games We now analyze a more general version of the game of hide-and-seek. Example 2.7.1 ( Generalized Hide-and-seek ) . A matrix of values ( b i,j ) n × n is given. Player II chooses a location ( i,j ) at which to hide. Player I chooses a row or a column of the matrix. He wins a payment of b i,j if the line he has chosen contains the hiding place of his opponent. First, we propose a strategy for player II, later checking that it is optimal. Player II first chooses a fixed permutation π of the set { 1 ,...,n } and then hides at location ( i,π i ) with a probability p i that he chooses. For example, of n = 5, and the fixed permutation π is 3 , 1 , 4 , 2 , 5, then the following matrix gives the probability of player II hiding in different places: 0 0 p 1 0 0 p 2 0 0 0 0 0 0 0 p 3 0 0 p 4 0 0 0 0 0 0 0 p 5 Given a permutation π , the optimal choice for p i is p i = d i,π i /D π , where d i,j = b 1 i,j and D π = n i =1 d i,π i , because it is this choice that equalizes the expected payments. For the fixed strategy, player I may choose to select
2.7 General hide-and-seek games 67 row i (for an expected payoff of p i b i,π ( i ) ) or column j (for an expected payoff of p j b π 1 ( j ) ,j ), so the expected payoff of the game is then max parenleftbigg max i p i b i,π ( i ) , max j p π 1 ( j ) b π 1 ( j ) ,j parenrightbigg = max parenleftbigg max i 1 D π , max j 1 D π parenrightbigg = 1 D π . Thus, if player II is going to use a strategy that consists of picking a permutation π and then doing as described, the right permutation to pick is one that maximizes D π . We will in fact show that doing this is an optimal strategy, not just in the restricted class of those involving permutations in this way, but over all possible strategies. To find an optimal strategy for player I, we need an analogue of K¨ onig’s lemma. In this context, a covering of the matrix D = ( d i,j ) n × n will be a pair of vectors u = ( u 1 ,...,u n ) and w = ( w 1 ,...,w n ), with non-negative components, such that u i + w j d i,j for each pair ( i,j ). The analogue of the K¨ onig lemma is Lemma 2.7.1. Consider a minimal covering ( u , w ) (i.e., one for which n i =1 ( u i + w i ) is minimal). Then n summationdisplay i =1 ( u i + w i ) = max π D π . (2.5) Proof. Note that a minimal covering exists, because the map ( u , w ) mapsto→ n summationdisplay i =1 ( u i + w i ) , defined on the closed and bounded set braceleftbig ( u , w ) : 0 u i ,w i M, and u i + w j d i,j bracerightbig , where M = max i,j d i,j , does indeed attain its infimum. Note also that we may assume that min i u i > 0.